Make the Cake: Preheat oven to 350 degrees F. Grease an 8-inch square baking pan.

In a small bowl, whisk together the flour, baking soda, and salt; set aside.

Beat the butter and sugar with an electric mixer on medium-high speed until pale and fluffy, about 3 minutes. Add the eggs, one at a time, and beat until combined. Add the bananas and vanilla extract and beat until incorporated.

Reduce the mixer speed to low and add the flour mixture in three additions, alternating with the milk in two additions, scraping down the bowl as needed. Using a rubber spatula, stir in the chocolate chips.

Transfer the batter to the prepared pan and smooth the top into an even layer. Bake until a toothpick inserted into the center comes out clean, 45 to 50 minutes, rotating the pan halfway through baking. Allow the cake to cool completely.

Make the Frosting: Using an electric mixer on medium speed, beat the butter and cream cheese on medium speed until smooth, 1 to 2 minutes. Reduce the mixer speed to low and add the powdered sugar; beat until thoroughly combined, then add the vanilla extract, increase the mixer speed to medium and beat until completely smooth, an additional 1 to 2 minutes.

Spread the frosting evenly over the top of the cake and serve. The cake can be stored, covered, at room temperature for up to 2 days, or in the refrigerator for up to 5 days.